QPR loanee Tom Carroll wants to "make it at Tottenham" but is prepared to consider all of his options at the end of the season.
Ilkay Gundogan has commited his future to Borussia Dortmund by signing a new contract.
The 21-year-old becomes Spurs' seventh signing of the summer, and follows Steaua Bucharest defender Vlad Chiriches into the club on a busy day.
Lamela started his career with River Plate in his homeland before joining Roma in 2011 for a fee of around £11million.
He scored four goals in his opening season in Serie A but made a massive breakthrough in 2012-13, when he hit the target 15 times in 33 games, second at the club behind recent Southampton signing Daniel Osvaldo.
Lamela has also won three international caps after progressing from their Under-20 set-up.
The fee paid for Lamela represents the third time this summer that Spurs have broken their transfer record, surpassing the £26million Spurs spent on Valencia striker Roberto Soldado earlier this month which in turn broke the mark they had set earlier in the summer when signing Brazil midfielder Paulinho.